318 Mr. W. Shanks on the Extension of [May 15, 



49. Our conclusion from the evidence before us is, that the residual 

 effect is a surface-effect more deeply seated than the gas-effect, but dis- 

 tributed outwards from the centre to the circumference, very much in the 

 same manner as the gas-effect. The residual effect likewise appears able 

 to penetrate a chamois-leather blind without any perceptible diminution. 

 W e regard these conclusions as prehminary, and shall endeavour in our 

 future experiments to procure additional evidence of these properties of 

 the residual effect, as well as to obtain new facts regarding it. In the 

 mean time, as the subject is one of interest, and has been already too long 

 delayed, we have not hesitated to bring these results before the notice of 

 the Royal Society. 



In concluding we would desire to express our thanks to Mr. F. Kingdon 

 for his assistance to us in many of these experiments. 



V. " On the Extension of the Numerical Value of it. V By Wil- 

 liam Shanks, Houghton-le- Spring, Durham. Communicated 

 by Prof. G. G. Stokes, Sec. U.S. Received April 16, 1873. 



In the ' Messenger of Mathematics ' for Dec. 1872, J. "W. L. Grlaisher, 

 Esq., has given some very interesting particulars regarding the calcula- 

 tion of ir, in the justness of which the author generally concurs. He, 

 however, differs from him as to the comparative merits of Van Ceulen, 

 who, in the early part of the seventeenth century, calculated 7r to 

 36 decimals. Huttou's formula also, given in the ' Messenger,' appears, 

 notwithstanding Hutton's own opinion, to be not so well adapted for ex- 

 tensive computation as Machin's, which the author has used on the pre- 

 sent as well as former occasions, regarding it as the best yet found. 



The values of tan -1 ± and of tan -1 -g-^-y are each given below to 709, 

 and the value of -k to 707 decimals. It will be observed that a few 

 figures in the values of tan -1 \ and of 7r, published in 1853, were erroneous. 

 The author detected the error quite recently, and has corrected it. The 



values of each term of the two series in ^=4 tan" 1 \ — tan -1 y-J-g, are far 



too bulky to be given in extenso : fortunately, but few would care to see 

 them ! 



It may here be stated that Prof. Richter, of Elbing, found 7r to 500 

 decimals in the year 1853 — all of which agree with the author's, published 

 early in the same year. 



The Society adjourned over Ascension Day to Thursday, May 29. 



